On the normal form of the Kirchhoff equation

TL;DR

This paper investigates the normal form of the Kirchhoff equation on a torus, showing that while cubic resonant terms are integrable, quintic resonant terms contribute non-trivially to energy estimates, with bounded transformations in Sobolev spaces.

Contribution

The paper performs a second step of normal form analysis for the Kirchhoff equation, revealing the non-integrable nature of quintic resonant terms.

Findings

Cubic resonant terms are integrable and do not affect energy estimates.

Quintic resonant terms contribute non-zero effects to energy estimates.

Normal form transformations are bounded in Sobolev spaces.

Abstract

Consider the Kirchhoff equation $$ \partial_{tt} u - \Delta u \Big( 1 + \int_{\mathbb{T}^d} |\nabla u|^2 \Big) = 0 $$ on the $d$-dimensional torus $\mathbb{T}^d$. In a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads to the question whether the same structure also emerges at the next steps of normal form. In this paper, we perform the second step and give a negative answer to the previous question: the quintic resonant terms give a nonzero contribution to the energy estimates. This is not only a formal calculation, as we prove that the normal form transformation is bounded between Sobolev spaces.